Quadratic Equation Solver

Step-by-step solutions, method comparison, and live parabola visualization in one workflow.

This solver evaluates each input with quadratic formula, factoring, and vertex-form logic, then identifies the route that fits the case. It reports real, repeated, complex, and linear boundary cases, with a live graph that shows how coefficient changes affect the curve and roots. For related equation workflows, explore algebra calculators.

Quick links: Method comparison | Worked examples | Check your result

Solve equations in the form:

$$ax^2 + bx + c = 0$$

Quick links: Live Chart | Quadratic Formula | Factoring | Vertex Form

If a = 0, the equation becomes linear (bx + c = 0).

Results

Solution type: — Discriminant: —

Roots

  • $x_1$: —
  • $x_2$: —

Quadratic curve and root behavior

What This Solver Handles

A quadratic equation can be written as $ax^2 + bx + c = 0$, where $a \ne 0$. Enter the coefficients in standard form to find the roots, classify the result, and inspect the corresponding parabola. The output separates two real roots, one repeated root, and a complex conjugate pair so you can see the type of result before relying on its decimal form.

The boundary case matters too. When $a = 0$, the expression is no longer quadratic. If $b \ne 0$, the solver treats it as a linear equation; if both $a$ and $b$ are zero, it reports that the input does not define a standard solvable quadratic. This prevents a formula from being applied outside its valid scope.

What the Discriminant Tells You

The discriminant is the part of the quadratic formula that determines the root type:

$$\Delta = b^2 - 4ac$$
  • $\Delta > 0$: two distinct real roots.
  • $\Delta = 0$: one repeated real root.
  • $\Delta < 0$: two complex conjugate roots.

Checking the discriminant first gives the rest of the calculation context. It tells you whether to expect two x-axis intersections, one point of contact, or no real intersection at all. The graph provides a visual check of that classification for the coefficients you entered.

Why This Solver Compares Methods

The quadratic formula provides a dependable route to roots for every valid quadratic. Factoring can be faster when the coefficient pattern is simple, while vertex form is useful when the position and shape of the parabola matter as much as the roots. The solver compares these approaches where they apply, rather than forcing every equation through the same explanation.

Use the method cards to see which routes are available for your input and why. A method that is not applicable is still useful information: it tells you something about the equation rather than leaving you to infer why a shortcut did not work.

Worked Quadratic Examples

Two Real Roots

For $x^2 - 5x + 6 = 0$, the discriminant is positive:

$$\Delta = (-5)^2 - 4(1)(6) = 1$$ $$x = \frac{5 \pm \sqrt{1}}{2}$$ $$x_1 = 3,\qquad x_2 = 2$$

The curve crosses the x-axis twice, which agrees with the two real roots.

A Repeated Root

For $x^2 - 4x + 4 = 0$, the discriminant is zero:

$$\Delta = (-4)^2 - 4(1)(4) = 0$$ $$x = \frac{4 \pm \sqrt{0}}{2} = 2$$

The parabola touches the x-axis at one point, so both root branches give the same value.

Complex Roots

For $x^2 + 2x + 5 = 0$, the discriminant is negative:

$$\Delta = 2^2 - 4(1)(5) = -16$$ $$x = \frac{-2 \pm \sqrt{-16}}{2} = -1 \pm 2i$$

There is no x-axis intersection because the roots are complex. The live graph confirms the same result by showing the curve entirely above the axis.

Check a Result Before You Use It

A short check catches most quadratic-solving mistakes:

  1. Rewrite the equation in standard form and preserve the signs of $a$, $b$, and $c$.
  2. Compute the discriminant before interpreting the roots.
  3. Keep the full denominator $2a$ when applying the formula.
  4. Substitute a reported real root into the original equation when the result will be reused.
  5. For complex roots, confirm that the two values form a conjugate pair.

The graph is a reasonableness check, not a replacement for the algebra. Use it to confirm the expected number of real intersections and to spot an input sign error before carrying a result into a calculation, report, or decision.

Explore more problem-solving tools in Math Calculators.